Draft version February 11, 2020

Typeset using LATEX modern style in AASTeX61

THE GREAT MARKARIAN 421 FLARE OF FEBRUARY 2010:

MULTIWAVELENGTH VARIABILITY AND CORRELATION STUDIES

A. U. Abeysekara,1 W. Benbow,2 R. Bird,3 A. Brill,4 R. Brose,5

M. Buchovecky,3 J. H. Buckley,6 J. L. Christiansen,7 A. J. Chromey,8

M. K. Daniel,2 J. Dumm,9 A. Falcone,10 Q. Feng,4 J. P. Finley,11

L. Fortson,9 A. Furniss,12 N. Galante,2 A. Gent,13 G. H. Gillanders,14

C. Giuri,15 O. Gueta,15 T. Hassan,15 O. Hervet,16 J. Holder,17 G. Hughes,2

T. B. Humensky,4 C. A. Johnson,16 P. Kaaret,18 P. Kar,1

N. Kelley-Hoskins,15 M. Kertzman,19 D. Kieda,1 M. Krause,15

F. Krennrich,8 S. Kumar,20 M. J. Lang,14 P. Moriarty,14 R. Mukherjee,21

T. Nelson,9 D. Nieto,4 M. Nievas-Rosillo,15 S. O’Brien,22 R. A. Ong,3

A. N. Otte,13 N. Park,23 A. Petrashyk,4 A. Pichel,24 M. Pohl,5

R. R. Prado,15 E. Pueschel,15 J. Quinn,22 K. Ragan,20 P. T. Reynolds,25

G. T. Richards,17 E. Roache,2 A. C. Rovero,24 C. Rulten,9 I. Sadeh,15

M. Santander,26 G. H. Sembroski,11 K. Shahinyan,9 B. Stevenson,3

I. Sushch,27 J. Tyler,20 V. V. Vassiliev,3 S. P. Wakely,28 A. Weinstein,8

R. M. Wells,8 P. Wilcox,18 A. Wilhelm,5 D. A. Williams,16 and B. Zitzer20

(VERITAS Collaboration)

V. A. Acciari,29 S. Ansoldi,30, 31 L. A. Antonelli,32 A. Arbet Engels,33

D. Baack,34 A. Babic,35 B. Banerjee,36 U. Barres de Almeida,37

J. A. Barrio,38 J. Becerra Gonzalez,29 W. Bednarek,39 L. Bellizzi,40

E. Bernardini,41, 42, 43 A. Berti,44, 45 J. Besenrieder,46 W. Bhattacharyya,41

C. Bigongiari,32 A. Biland,33 O. Blanch,47 G. Bonnoli,40 G. Busetto,42

R. Carosi,48 G. Ceribella,46 Y. Chai,46 S. Cikota,35 S. M. Colak,47

U. Colin,46 E. Colombo,29 J. L. Contreras,38 J. Cortina,47 S. Covino,32

V. D’Elia,32 P. Da Vela,48 F. Dazzi,32 A. De Angelis,42 B. De Lotto,30

M. Delfino,47, 49 J. Delgado,47, 49 F. Di Pierro,44 E. Do Souto Espinera,47

D. Dominis Prester,35 D. Dorner,50 M. Doro,42 S. Einecke,34 D. Elsaesser,34

V. Fallah Ramazani,51 A. Fattorini,34 A. Fernandez-Barral,42

G. Ferrara,32 D. Fidalgo,38 L. Foffano,42 M. V. Fonseca,38 L. Font,52

C. Fruck,46 D. Galindo,53 S. Gallozzi,32 R. J. Garcıa Lopez,29

M. Garczarczyk,41 S. Gasparyan,54 M. Gaug,52 N. Godinovic,35 D. Green,46

D. Guberman,47 D. Hadasch,31 A. Hahn,46 J. Herrera,29 J. Hoang,38

D. Hrupec,35 S. Inoue,31 K. Ishio,46 Y. Iwamura,31 H. Kubo,31 J. Kushida,31

A. Lamastra,32 D. Lelas,35 F. Leone,32 E. Lindfors,51 S. Lombardi,32

F. Longo,30, 45 M. Lopez,38 R. Lopez-Coto,42 A. Lopez-Oramas,29

B. Machado de Oliveira Fraga,37 C. Maggio,52 P. Majumdar,36

M. Makariev,55 M. Mallamaci,42 G. Maneva,55 M. Manganaro,35

K. Mannheim,50 L. Maraschi,32 M. Mariotti,42 M. Martınez,47 S. Masuda,31

D. Mazin,46, 31 D. Miceli,30 M. Minev,55 J. M. Miranda,40 R. Mirzoyan,46

Corresponding author: Lucy Fortsonlfortson@umn.edu

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E. Molina,53 A. Moralejo,47 D. Morcuende,38 V. Moreno,52 E. Moretti,47

P. Munar-Adrover,52 V. Neustroev,51 A. Niedzwiecki,39

M. Nievas Rosillo,38 C. Nigro,41 K. Nilsson,51 D. Ninci,47 K. Nishijima,31

K. Noda,31 L. Nogues,47 M. Nothe,34 S. Paiano,42 J. Palacio,47

M. Palatiello,30 D. Paneque,46 R. Paoletti,40 J. M. Paredes,53 P. Penil,38

M. Peresano,30 M. Persic,30, 56 P. G. Prada Moroni,48 E. Prandini,42

I. Puljak,35 W. Rhode,34 M. Ribo,53 J. Rico,47 C. Righi,32 A. Rugliancich,48

L. Saha,38 N. Sahakyan,54 T. Saito,31 K. Satalecka,41 T. Schweizer,46

J. Sitarek,39 I. Snidaric,35 D. Sobczynska,39 A. Somero,29 A. Stamerra,32

D. Strom,46 M. Strzys,46 S. Sun,57 T. Suric,35 F. Tavecchio,32 P. Temnikov,55

T. Terzic,35 M. Teshima,46, 31 N. Torres-Alba,53 S. Tsujimoto,31

J. van Scherpenberg,46 G. Vanzo,29 M. Vazquez Acosta,29 I. Vovk,46

M. Will,46 and D. Zaric35

(MAGIC Collaboration)

H. D. Aller,58 M. F. Aller,58 M. T. Carini,59 D. Horan,60 B. Jordan,61

Svetlana G. Jorstad,62, 63 O.M. Kurtanidze,64, 65, 66 S.O. Kurtanidze,64

A. Lahteenmaki,67, 68 V. M. Larionov,63, 69 E. G. Larionova,63 G. Madejski,70

Alan P. Marscher,62 W. Max-Moerbeck,71 J. Ward Moody,72

D. A. Morozova,63 M.G. Nikolashvili,64 C. M. Raiteri,73

A. C. S. Readhead,74 J. L. Richards,11 Alberto C. Sadun,75 T. Sakamoto,76

L.A. Sigua,64 P. S. Smith,77 H. Talvikki,78 J. Tammi,67 M. Tornikoski,67

I. S. Troitsky,63 and M. Villata73

(Multiwavelength Partners)

1Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA2Center for Astrophysics — Harvard & Smithsonian, Fred Lawrence Whipple Observatory, Amado,

AZ 85645, USA3Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA4Physics Department, Columbia University, New York, NY 10027, USA5Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany and

DESY, Platanenallee 6, 15738 Zeuthen, Germany6Department of Physics, Washington University, St. Louis, MO 63130, USA7Physics Department, California Polytechnic State University, San Luis Obispo, CA 94307, USA8Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA9School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA10Department of Astronomy and Astrophysics, 525 Davey Lab, Pennsylvania State University,

University Park, PA 16802, USA11Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA12Department of Physics, California State University - East Bay, Hayward, CA 94542, USA13School of Physics and Center for Relativistic Astrophysics, Georgia Institute of Technology, 837

State Street NW, Atlanta, GA 30332-043014School of Physics, National University of Ireland Galway, University Road, Galway, Ireland15DESY, Platanenallee 6, 15738 Zeuthen, Germany

3

16Santa Cruz Institute for Particle Physics and Department of Physics, University of California,

Santa Cruz, CA 95064, USA17Department of Physics and Astronomy and the Bartol Research Institute, University of Delaware,

Newark, DE 19716, USA18Department of Physics and Astronomy, University of Iowa, Van Allen Hall, Iowa City, IA 52242,

USA19Department of Physics and Astronomy, DePauw University, Greencastle, IN 46135-0037, USA20Physics Department, McGill University, Montreal, QC H3A 2T8, Canada21Department of Physics and Astronomy, Barnard College, Columbia University, NY 10027, USA22School of Physics, University College Dublin, Belfield, Dublin 4, Ireland23WIPAC and Department of Physics, University of Wisconsin-Madison, Madison WI, USA24Instituto de Astronoma y Fsica del Espacio (IAFE, CONICET-UBA), CC 67 - Suc. 28,

(C1428ZAA) Ciudad Autnoma de Buenos Aires, Argentina25Department of Physical Sciences, Cork Institute of Technology, Bishopstown, Cork, Ireland26Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA27Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany28Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA29Inst. de Astrofısica de Canarias, E-38200 La Laguna, and Universidad de La Laguna, Dpto.

Astrofısica, E-38206 La Laguna, Tenerife, Spain30Universita di Udine, and INFN Trieste, I-33100 Udine, Italy31Japanese MAGIC Consortium: ICRR, The University of Tokyo, 277-8582 Chiba, Japan;

Department of Physics, Kyoto University, 606-8502 Kyoto, Japan; Tokai University, 259-1292

Kanagawa, Japan; RIKEN, 351-0198 Saitama, Japan32National Institute for Astrophysics (INAF), I-00136 Rome, Italy33ETH Zurich, CH-8093 Zurich, Switzerland34Technische Universitat Dortmund, D-44221 Dortmund, Germany35Croatian MAGIC Consortium: University of Rijeka, 51000 Rijeka; University of Split - FESB,

21000 Split; University of Zagreb - FER, 10000 Zagreb; University of Osijek, 31000 Osijek; Rudjer

Boskovic Institute, 10000 Zagreb, Croatia36Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Salt Lake, Sector-1, Kolkata 700064,

India37Centro Brasileiro de Pesquisas Fısicas (CBPF), 22290-180 URCA, Rio de Janeiro (RJ), Brasil38Unidad de Partıculas y Cosmologıa (UPARCOS), Universidad Complutense, E-28040 Madrid,

Spain39University of Lodz, Department of Astrophysics, PL-90236 Lodz, Poland40Universita di Siena and INFN Pisa, I-53100 Siena, Italy41Deutsches Elektronen-Synchrotron (DESY), D-15738 Zeuthen, Germany42Universita di Padova and INFN, I-35131 Padova, Italy43Humboldt University of Berlin, Institut fur Physik D-12489 Berlin Germany44Istituto Nazionale Fisica Nucleare (INFN), 00044 Frascati (Roma) Italy45Dipartimento di Fisica, Universita di Trieste, I-34127 Trieste, Italy46Max-Planck-Institut fur Physik, D-80805 Munchen, Germany

4

47Institut de Fısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology

(BIST), E-08193 Bellaterra (Barcelona), Spain48Universita di Pisa, and INFN Pisa, I-56126 Pisa, Italy49Port d’Informacio Cientıfica (PIC) E-08193 Bellaterra (Barcelona) Spain50Universitat Wurzburg, D-97074 Wurzburg, Germany51Finnish MAGIC Consortium: Tuorla Observatory (Department of Physics and Astronomy) and

Finnish Centre of Astronomy with ESO (FINCA), University of Turku, FI-20014 Turku, Finland;

Astronomy Division, University of Oulu, FI-90014 Oulu, Finland52Departament de Fısica, and CERES-IEEC, Universitat Autonoma de Barcelona, E-08193

Bellaterra, Spain53Universitat de Barcelona, ICCUB, IEEC-UB, E-08028 Barcelona, Spain54ICRANet-Armenia at NAS RA, 0019 Yerevan, Armenia55Inst. for Nucl. Research and Nucl. Energy, Bulgarian Academy of Sciences, BG-1784 Sofia,

Bulgaria56INAF-Trieste and Dept. of Physics & Astronomy, University of Bologna57Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 2005

Songhu Road, Shanghai 200438, China58Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107, USA59Department of Physics and Astronomy, Western Kentucky University, 1906 College Heights

Boulevard #11077, Bowling Green, KY 42101, USA60Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France61School of Cosmic Physics, Dublin Institute For Advanced Studies, Ireland62Institute for Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA

0221563Astronomical Institute, St. Petersburg State University, Universitetskij Pr. 28, Petrodvorets,

198504 St. Petersburg, Russia64Abastumani Observatory, Mt. Kanobili, 0301 Abastumani, Georgia65Engelhardt Astronomical Observatory, Kazan Federal University, Tatarstan, Russia66Center for Astrophysics, Guangzhou University, Guangzhou 510006, China67Aalto University Metsahovi Radio Observatory, Metsahovintie 114, FI-02540 Kylmala, Finland68Aalto University Department of Electronics and Nanoengineering, P.O. BOX 15500, FI-00076

AALTO, Finland69Pulkovo Observatory, St. Petersburg, Russia70W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and

Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stanford University,

Stanford, CA 94305, USA71Universidad de Chile, Departamento de Astronoma, Camino El Observatorio 1515, Las Condes,

Santiago, Chile72Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA73INAF, Osservatorio Astrofisico di Torino, I-10025 Pino Torinese (TO), Italy74Cahill Centre for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA

91125, USA75Department of Physics, University of Colorado Denver, Denver, Colorado, CO 80217-3364, USA

5

76Department of Physics and Mathematics, College of Science and Engineering, Aoyama Gakuin

University, 5-10-1 Fuchinobe,Chuoku, Sagamihara-shi Kanagawa 252-5258, Japan77Steward Observatory, University of Arizona, Tucson, AZ 85721 USA78Tuorla Observatory, Department of Physics and Astronomy, University of Turku, 20014 Turku,

Finland

Submitted to ApJ

ABSTRACT

We report on variability and correlation studies using multiwavelength observations

of the blazar Mrk 421 during the month of February, 2010 when an extraordinary

flare reaching a level of ∼27 Crab Units above 1 TeV was measured in very-high-

energy (VHE) γ-rays with the VERITAS observatory. This is the highest flux state

for Mrk 421 ever observed in VHE γ-rays. Data are analyzed from a coordinated

campaign across multiple instruments including VHE γ-ray (VERITAS, MAGIC),

high-energy (HE) γ-ray (Fermi -LAT), X-ray (Swift, RXTE, MAXI), optical (includ-

ing the GASP-WEBT collaboration and polarization data) and radio (Metsahovi,

OVRO, UMRAO). Light curves are produced spanning multiple days before and af-

ter the peak of the VHE flare, including over several flare ‘decline’ epochs. The main

flare statistics allow 2-minute time bins to be constructed in both the VHE and opti-

cal bands enabling a cross-correlation analysis that shows evidence for an optical lag

of ∼25–55 minutes, the first time-lagged correlation between these bands reported on

such short timescales. Limits on the Doppler factor (δ & 33) and the size of the emis-

sion region (δ−1RB . 3.8 × 1013 cm) are obtained from the fast variability observed

by VERITAS during the main flare. Analysis of 10-minute-binned VHE and X-ray

data over the decline epochs shows an extraordinary range of behavior in the flux-flux

relationship: from linear to quadratic to lack of correlation to anti-correlation. Taken

together, these detailed observations of an unprecedented flare seen in Mrk 421 are

difficult to explain by the classic single-zone synchrotron self-Compton model.

Keywords: galaxies: jets — galaxies: BL Lacertae objects: individual

(Mrk 421) — gamma rays: observations — X-rays: galaxies

6

1. INTRODUCTION

Blazars are a sub-class of radio-loud active galactic nuclei (AGN) with jets of rela-

tivistic material beamed nearly along the line-of-sight (Blandford & Rees 1978; Urry

& Padovani 1995) whose non-thermal radiation is observed across the entire spectrum,

from radio to γ-rays. Due to Doppler beaming, the bolometric luminosity of blazars

can be dominated by very-high-energy (VHE; > 100 GeV) γ-rays. At a redshift of

z=0.031, Mrk 421 is the closest known BL Lac object (de Vaucouleurs et al. 1995)

and is the first extragalactic object to be detected in VHE γ-rays (Punch et al. 1992).

Blazars now comprise the majority source class of VHE extragalactic γ-ray emitters

(Wakely & Horan 2008) and while there is much we have learned from multiwave-

length data taken over the past forty years on Mrk 421 and other blazars, there remain

many unanswered questions. Indeed, there is still no general agreement on the particle

acceleration mechanism within the jet or the location of γ-ray emission zone(s) (e.g.,

Boettcher 2019). Nonetheless, progress can be made through dedicated campaigns

organized simultaneously across as many wavebands as possible (e.g., Aleksic et al.

2015a; Furniss et al. 2015; Ahnen et al. 2018).

The spectral energy distribution (SED) of blazars is characterized by a double peak

where the lower peak is due to synchrotron radiation while the higher peak is generally

thought to arise from inverse-Compton (IC) upscattering of lower-energy photons off

the population of accelerating electrons in the jet (Jones et al. 1974). Hadronic models

(Aharonian 2000; Mucke & Protheroe 2001; Mannheim 1993; Dimitrakoudis et al.

2014) or even lepto-hadronic models (Cerruti et al. 2015), may also be responsible for

the second SED peak. The Synchrotron-Self Compton (SSC) model posits that the

seed photons for the IC process are the synchrotron photons from the accelerating

electrons (e.g., Ghisellini et al. 1998). Observationally, blazars are classified by the

peak frequency of their synchrotron emission; with νs = 1018.9 Hz, Mrk 421 is deemed

a high-frequency peaked BL Lac (HBL) (Nieppola et al. 2006).

Blazars exhibit complex temporal structures with strong variability across the spec-

trum from radio to γ-rays (e.g. Romero et al. (2017) and references therein). Blazar

light curves are typically aperiodic with power-law Power Spectral Density (PSD)

distributions indicative of stochastic processes (Finke & Becker 2015). Multi-band

blazar light curves can be punctuated by dramatic flares on timescales from minutes

to days where inter-band correlation is often observed (Acciari et al. 2011b; H. E. S. S.

Collaboration et al. 2012; Ahnen et al. 2018).

Studying the time-varying characteristics of a source through multiwavelength cam-

paigns can test model predictions on what governs the γ-ray emission and its location

within the jet. The standard homogeneous single-zone SSC model of blazar emission

employs a single population of electrons that is accelerated in a compact region < 1

pc from the central engine (the central black hole driving the jet). The accelerated

electrons cool through the emission of synchrotron radiation, then potentially through

IC scattering and/or escape out of the accelerating “blob”. The spatial scale of the

7

emission region can be set by the variability detected in the VHE-band observations.

Competition between cooling, acceleration and dynamical timescales that character-

ize the system can lead to several potential observables including asymmetries in flare

profiles and “soft” or “hard” lags (and accompanying clockwise or counter-clockwise

hysteresis loops) as described in e.g. Kirk et al. (1998) or Li & Kusunose (2000).

Much of the previous work with Mrk 421 as well as the ever-growing population

of blazars detected by VHE instruments indicates that most SEDs of HBLs can be

described by a single-zone SSC model (Acciari et al. 2011a; Abeysekara et al. 2017;

Ahnen et al. 2018). As tracers of the same underlying electron population, hard X-

rays typically probe the falling edge of the synchrotron peak while VHE γ-rays probe

the falling edge of the IC peak in an HBL with the expectation that these bands will

show highly correlated fast variability. However, orphan flares, such as the 2002 VHE

flare observed in 1ES 1959+650 (Krawczynski et al. 2004) without a corresponding

X-ray flare, provide evidence that one-zone SSC models are too simplistic. The re-

markable VHE flare in PKS 2155-304 seen by the High Energy Stereoscopic System

(H.E.S.S.) in late July, 2006 suggests a need for two emission zones to explain the

data (Aharonian et al. 2009). Several recent campaigns on Mrk 421 and Mrk 501 also

indicate a preference for a multi-component scenario (Aleksic et al. 2015b; Ahnen

et al. 2017a).

Fast flaring events provide another test of the SSC model. Several blazars have been

observed to emit VHE flares that vary on timescales of 5-20 minutes (Albert et al.

2007; Aharonian et al. 2009). Mrk 421 itself has a history of fast flares including those

reported in Gaidos et al. (1996); B lazejowski et al. (2005); Fossati et al. (2008); Acciari

et al. (2011a). These ∼minute timescale flares pose serious issues for single-zone

blazar models as the implied high bulk Lorentz factors required are in tension with

the radio observations of these sources (Bottcher et al. 2013; Piner & Edwards 2018).

Moreover, the shock-in-jet model suggested to explain knots of material traveling

along the jet in radio observations is found to be incompatible with the highly-compact

emission regions implied by fast flaring episodes detected in blazars (Romero et al.

2017). Indeed, since the majority of blazars are detected during flaring episodes, the

erroneous interpretation could be made that a single-zone SSC scenario is responsible

for the generic form of the object’s SED. In fact, there may be more than one emitting

region at any given time, with one region accounting for “quiescent” or “envelope”

behavior while another region or process may be responsible for a detected flare

triggered by a localized event (e.g. magnetic reconnection (Petropoulou et al. 2016)).

Given the sometimes surprising and dynamical nature of blazars, efforts to coor-

dinate multiwavelength campaigns continue to be important. The results from each

campaign provide further clues for modelers to incorporate. For example, highly

correlated rapid variability observed between the VHE and optical bands such as de-

scribed in this work, has not been reported before, and has not been accounted for

in modeling. The observed (or lack of observed) correlated activity between specific

8

bands can discriminate between possible emission mechanisms; and stringent con-

straints on the sizes and locations of γ-ray emission regions can be set by the flux

and spectral variability patterns of blazars (Boettcher 2012).

In this paper, we apply timing analysis techniques, including variability and cor-

relation studies, to the extraordinary Mrk 421 flare recorded in February, 2010 by

the VERITAS observatory and many multiwavelength partners. During 2009-2010,

Mrk 421 was the object of an intense multiwavelength campaign organized by the

Fermi Large Area Telescope (Fermi -LAT) collaboration and involving the ground-

based imaging air-Cherenkov telescopes (IACTs) (H.E.S.S., MAGIC and VERITAS)

as well as RXTE and Swift satellites in X-ray, Swift UVOT ultraviolet, and numer-

ous ground-based optical and radio telescopes. Several smaller flares were observed

throughout the campaign including one in March, 2010 described in (Aleksic et al.

2015b). Here we report on the multiwavelength dataset covering the period 1 Febru-

ary - 1 March, 2010 UT (MJD 55228 - 55256) with a focus on the giant VHE flare

on 17 February, 2010 UT (MJD 55244). We note that several other instruments have

observed the same flare including MAXI (Isobe et al. 2010), H.E.S.S. (Tluczykont

et al. 2011), HAGAR (Shukla et al. 2012) and TACTIC (Singh et al. 2015).

This paper is organized as follows: in Section §2 we describe the multiwavelength

datasets including the respective methods for analyzing the data presented. In Sec-

tion §3 we focus on the results from the night of the exceptional flare including the

variability analysis of VERITAS data as well as results from optical-VHE correlation

studies. The results of further multiwavelength studies over the full February 2010

dataset are presented in Section §4, including multiwavelength variability studies as

well as VHE-X-ray and HE-X-ray correlation analyses. We conclude with an overall

discussion of the results in Section §5.

2. DATASETS AND DATA REDUCTION

The multiwavelength light curves covering radio-to-VHE observations around the

time of the Mrk 421 February, 2010 flare are shown in Figure 1. While the light curves

in Figure 1 are meant as an overview of available observations, they demonstrate the

full breadth of the campaign and show the progression of the flare; more detailed light

curves in the various wavebands are considered later in the paper. We summarize the

available datasets and present details of the instruments in the following subsections.

The data for light curves used throughout this paper are available through the on-line

version of the article.

2.1. VHE γ-ray observations

The VHE γ-ray data comprise both MAGIC and VERITAS observations. Starting

with the MAGIC observatory, data were taken on Mrk 421 between MJD 55234 and

55240 (7 February and 13 February, 2010) with bad weather preventing further ob-

servations. Upon alert from the campaign that the X-ray state was quite high and

variable, VERITAS picked up the observations between MJD 55244 and 55247 (17

9

0.51.01.5

> 20

0 Ge

V Fl

ux[ 1

09 c

m2 s

1 ]

VERITASMAGIC

0.0

5.0

0.1-100 GeV Flux[ 10

7cm2 s

1]

Fermi-LAT

0.0

0.5

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keV

Coun

t Rat

e [ s

1 ]

MAXI

0.0

2.0

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9 erg cm2 s

1 ]

Swift-XRTRXTE

1.251.501.75

Flux

[ 1

010

erg

cm

2 s1

UVOT uvw2UVOT uvm2UVOT uvw1

152025

Flux Density[ m

Jy ]

R-band

0.50

0.75

Flux

Den

sity

[ Jy

] Radio

55230 55235 55240 55245 55250 55255Date [ MJD ]

0.0

5.0

p%

500-700 nm p%

0

100

EVPA[ deg ]

500-700 nm EVPA

Figure 1. Light curves for multi-band observations during the February, 2010 portionof the Fermi -LAT led campaign. From top to bottom: VHE (VERITAS, MAGIC), HE(Fermi -LAT), hard X-ray (MAXI), X-ray (RXTE, Swift-XRT), UV (Swift-UVOT), opti-cal (Abastumani (blue), CRAO (orange), GRT (green), GalaxyView (red), KVA (purple),NewMexicoSkies (brown), Perkins (pink), RCT (gray), and Steward (tan)), and radio (UM-RAO 8 GHz (blue stars) and 14 GHz (orange squares), OVRO 15 GHz (green circles) andMetsahovi 37 GHz (red triangles)), including optical polarization observations (StewardObservatory). The light curves are binned by individual observation except for VHE, HE,and MAXI which are daily binned. The time of the giant VHE flare is denoted by thedashed vertical line. The dotted horizontal line on the VHE (top panel) denotes 1 CrabUnit based on Aharonian et al. (2006). Note that error bars are not visible for several bandsdue to high statistics.

February and 20 February, 2010). As soon as VERITAS began taking Mrk 421 data

on MJD 55244, VERITAS observed a remarkable flare in progress with a peak flux

∼ 15 Crab Units (CU) above 200 GeV (CU based on Aharonian et al. 2006). Over

the next two days, VERITAS observed the flux decrease to the average over the 2009-

2010 season, ∼ 1 CU, which was itself an elevated state (see Acciari et al. 2011a).

The MAGIC and VERITAS combined VHE γ-ray light curve for Mrk 421 is shown in

the top panel of Figure 1 in daily bins for data taken between MJD 55234 and 55247.

10

The VHE data with considerably finer binning are described below in Sections §3.1,

§3.3 and §4.3.

VERITAS. —The VERITAS array (Holder et al. 2006; Acciari et al. 2008) is located

at 1300 m above sea level in Arizona at the Fred Lawrence Whipple Observatory (31◦

40′ N, 110◦ 57′ W) and comprises four Davies-Cotton-design telescopes each with a

12-m diameter primary reflector. During the observations presented here, VERITAS

was sensitive to γ-rays between 100 GeV and tens of TeV with an energy resolution

better than ∼20%, and an integral flux sensitivity that would have allowed a point

source detection with a 1% Crab Nebula flux in less than 30 hours.

A total of 17 hours 21 minutes of Mrk 421 data were taken by VERITAS over the

month of February of which 16 hours 44 minutes were data taken in good weather

conditions. A total of 5 hours 12 minutes of data were collected on the night of the

giant flare (MJD 55244), with observations starting at 83◦ elevation and ending with

an elevation of 40◦, giving rise to a higher-energy threshold for later observations.

Three of the four telescopes were operational during this time (on MJD 55244). All

four telescopes participated in the rest of the February data and all observations were

made in wobble mode (Fomin et al. 1994). Data were analyzed and cross-checked

with the two standard VERITAS analysis packages (Cogan 2008; Daniel 2008).

MAGIC. —The Major Atmospheric Gamma-ray Imaging Cherenkov (MAGIC) tele-

scope system consists of two telescopes each with a 17-m diameter mirror dish, located

at 2200 m above sea level at the Roque de los Muchachos, on the Canary Island of

La Palma ( 28◦ 46′ N, 18◦ 53′ W) (Albert et al. 2008; Aleksic et al. 2012, 2016).

The MAGIC data for Mrk 421 in February, 2010 comprise a total of 2 hours over

four separate observations. The data were taken in wobble mode at an elevation

above 60◦ to achieve the lowest-possible energy threshold. These data were analyzed

following the standard procedure Aleksic et al. (2012) with the MAGIC Analysis and

Reconstruction Software (MARS Moralejo et al. 2009).

2.2. HE γ-ray observations

The Large Area Telescope (LAT) aboard the Fermi Gamma-Ray Space Telescope

(FGST) is a pair-conversion detector sensitive to γ-rays between 20 MeV and 300

GeV. FGST typically operates in survey mode such that Mrk 421 is observed once

every ∼ 3 hours (Atwood et al. 2009).

Events belonging to the Source data class with energies between 100 MeV and

100 GeV were selected and analyzed using the P8R2 SOURCE V6 instrument response

functions and v10r0p5 of the Fermi ScienceTools.1 In order to avoid contamination

from Earth limb photons, a zenith angle cut of < 90◦ was applied. The analysis

considered data from MJD 55230 to 55255, which is the 25-day period centered on

the peak of the TeV flare as detected by VERITAS.

1 http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/

11

The full dataset was analyzed using a binned likelihood analysis. The likelihood

model included all Fermi -LAT sources from the third Fermi catalog (Acero et al.

2015) located within a 15◦ region-of-interest (RoI) centered on Mrk 421, as well as

the isotropic and Galactic diffuse emission. For the full dataset, Mrk 421 was fitted

with a power-law model, with both the flux normalization and photon index being left

as free parameters in the likelihood fit. All spectral parameters were fixed for sources

located at > 7◦, while the normalization parameter was fitted for sources between 3◦

and 7◦, and all parameters were fitted for sources < 3◦ from the RoI center.

The optimized RoI model from the full dataset was used to calculate the Mrk 421

daily-binned light curve. The likelihood analysis was repeated for each time bin to

obtain the daily flux points. Only the normalization parameter for Mrk 421 was fitted,

while all other RoI model parameters were kept fixed. The resulting Mrk 421 light

curve with daily binning is shown in the second panel from the top in Figure 1. Fitting

the index parameter of the Mrk 421 model along with the normalization parameter

has an insignificant impact on the result.

2.3. X-ray observations

We obtained X-ray data over the period of interest from three different observatories:

MAXI, the Rossi X-ray Timing Experiment (RXTE ) and Swift. Just prior to the

observed TeV flare, a flare in both HE (Fermi -LAT, Abdo et al. 2011) and X-ray

(MAXI, Isobe et al. 2010) was observed (without simultaneous VHE observations).

This HE/X-ray flare triggered the VHE observations.

MAXI. —MAXI is an all-sky monitoring instrument onboard the International Space

Station, and is sensitive to X-rays in the energy range 0.5–30 keV (Matsuoka et al.

2009). We downloaded the daily-binned light curve for the entire month of February

from the MAXI Science Center data archive2. Mrk 421 is bright enough to result in

a significant detection in each 24-hour time bin. The resulting light curve, presenting

the 2–20 keV count rate in daily time bins, is shown in the third panel from the top

in Figure 1.

RXTE-PCA. —We observed Mrk 421 with the Proportional Counting Array (PCA)

instrument onboard RXTE through two observing programs (Obs IDs: 95386, 95133).

A total of 42 RXTE observations were carried out between 1 February, 2010 and 1

March, 2010. The RXTE datasets relevant for this paper are shown in the fourth

panel from the top in Figure 1 binned by individual observations.

For each observation, we extracted the spectrum from the Standard-2, binned-mode

data (i.e. 129 channel spectra accumulated every 16 seconds) using HEASoft v6.11. We

screened the data so that the angular separation between Mrk 421 and the pointing

direction was less than 0.05◦, the elevation angle was greater than 5◦, the time since

the last passage of the South Atlantic Anomaly was greater than 25 minutes, and the

2 http://maxi.riken.jp

12

electron contamination was low (ELECTRON2 < 0.1). We estimated the background

using the L7 model for Epoch 5C—the Proportional Counting Unit (PCU) count rate

of Mrk 421 was close to the transition point where the bright background model is

recommended over the faint background (40 cts/s/PCU). We chose the background

model based on the observed mean count rate in each observation. All spectra were

accumulated from both anodes in the upper Xenon layer of PCU2, which is turned on

in every observation and was the only PCU in operation in most of our observations.

Since the PCA has low sensitivity below 2.5 keV, and Mrk 421 is faint above 20 keV,

we analyzed the background-subtracted spectra in the energy range 2.5-20 keV.

To enable a more careful study of the X-ray behavior as well as joint X-ray/VHE

γ-ray behavior during the decline phases described in Section 4.3, we produced more-

detailed light curves for observations taken during the P95133 period (see Figures 6

and 8). We determined the RXTE count rate with the REX analysis tool3 using the

same extraction criterion as above. Light curves were first extracted in 16 second

time bins and then re-binned using the ftool lcurve to create 10-minute time bins.

Swift-XRT. —We analyzed Swift observations of Mrk 421 from two observing pro-

grams: 31630 and 30352 (the latter initiated in response to the VHE flare). A total

of 23 observations were carried out between 1 February, 2010 and 1 March, 2010. The

light curve from these observations is shown in the fourth panel from the top in Figure

1 binned by individual observations. Due to the high count rate of the source (>20

cts/s) all observations were obtained in windowed timing (WT) mode. We reran the

Swift data reduction pipeline on all datasets (xrtpipeline v0.12.6) to produce cleaned

event files and exposure maps. We created source spectra using XSelect v2.4b, ex-

tracting source events from a circular region of radius 40′′ centered on the source. We

subsequently created ancillary response files using xrtmkarf v0.5.9, applying a PSF

and dead-pixel correction using the exposure map created with xrtpipeline. Finally,

the appropriate response matrix file (in this case swxwt0to2s6 20010101v013.rmf) was

taken from the Swift calibration database. We grouped the spectra to have a mini-

mum of 20 counts per bin in order to facilitate the use of χ2-statistics in Xspec and

carried out model fits in the 0.3–10 keV energy range.

2.4. Optical Observations

UVOT. —The Ultraviolet/Optical Telescope onboard Swift also obtained data during

each observation, in one of three UV filters (UVW2, UVM2 or UVW1) for a total

of 59 exposures. All of the data taken between 7 February, 2010 and 20 February,

2010 were analyzed and are shown in the fifth panel from the top of Figure 1 binned

by individual observations. After extracting the source counts from an aperture of

5.0′′ radius around Mrk 421 and the background counts from four neighboring regions,

each of the same size, the magnitudes were computed using the uvotsource4 tool.

3 http://heasarc.gsfc.nasa.gov/docs/xte/recipes/rex.html)4 HEASOFT v6.13, Swift Rel4.0(Bld29) 14Dec2012 with calibrations from Breeveld et al. (2011).

13

These were converted to fluxes using the central wavelength values for each filter

from Poole et al. (2008). The observed fluxes were corrected for Galactic extinction

following the procedure and Rv value in Roming et al. (2009). An E(B−V ) value of

0.013 from Schlafly & Finkbeiner (2011) was used.

Ground-based Optical Observatories. —The optical fluxes reported in this paper were

obtained within the GASP-WEBT program (e.g. Villata et al. 2008, 2009), with

the optical telescopes at Abastumani, Roque de los Muchachos (KVA), Crimean,

and Lowell (Perkins) observatories. Additional observations were performed with

the Goddard Robotic Telescope (GRT), Galaxy View, and New Mexico Skies. All

instruments used the calibration stars reported in Villata et al. (1998) for calibration.

The Galactic extinction was corrected with the reddening corrections given in Schlegel

et al. (1998). The flux from the host galaxy (which is significant only below ν ∼1015 Hz) was estimated using the flux values across the R band from Nilsson et al.

(2007) and the colors reported in Fukugita et al. (1995), and subsequently subtracted

from the measured flux. The flux values obtained by the various observatories during

the same 6 hour time period agree within uncertainties except for the GRT, which

shows a flux systematically 15% lower than that of the other telescopes. We therefore

assume this represents a systematic error in the data, and correct the observed fluxes

to match the fluxes from the other observatories during the same 6-hour time interval.

Additionally, high-cadence, 2-minute exposure optical R-band observations near-

simultaneous with VERITAS were obtained on 17 February, 2010 with the 1.3-m

Robotically Controlled Telescope (RCT) located at Kitt Peak National Observatory.

The RCT observations started ∼50 minutes after the beginning of the VERITAS

observations and ended ∼15 minutes after VERITAS stopped observing.

The reported fluxes from all optical observatories include instrument-specific offsets

of a few mJy. These are due to differences in filter spectral responses and analysis

procedures of the various optical data sets combined with the host-galaxy contribution

(about 1/3 of the total flux measured for Mrk 421 in the R band). The following

offsets were determined and corrected for by using simultaneous observations and

treating several of the GASP-WEBT instruments as reference: GRT = 2.5 mJy;

RCT = -1.0 mJy; CRAO = 3.0 mJy; RCT for the long observations on 17 February

= 4.0 mJy. Moreover, a point-wise fluctuation of 0.2 mJy (∼0.01 mag) was added in

quadrature to the statistical uncertainties in order to account for potential day-to-day

differences for observations with the same instrument.

The reconstructed optical fluxes are shown in the sixth panel from the top of Figure

1 binned by individual observations. The 2-minute-binned VERITAS and RCT light

curves are displayed in Figure 4; these latter light curves are used in the discrete

cross-correlation analysis detailed in Section 3.3.

Steward Observatory - Optical Polarization. —Optical observations of Mrk 421 were

made during the high-energy monitoring campaign by the Steward Observatory (SO)

14

2.3m Bok Telescope on Kitt Peak, Arizona. The source was observed on seven consec-

utive nights from 13 February, 2010 (MJD 55240) through 19 February, 2010 (MJD

55246) using the SPOL imaging/spectropolarimeter (Schmidt et al. 1992). On all

seven nights, flux and polarization spectra spanning 4000-7550 A were acquired using

a 600 lines/mm grating in first order, which gives a dispersion of 4 A pixel−1. The po-

larization measurements employed a 3 arcsec-wide slit, yielding a resolution of ∼20 A.

The slit length was 51′′, long enough to sample the sky background from a region with-

out a significant amount of light from the host elliptical galaxy of Mrk 421 (Ulrich

et al. 1975). Data reduction followed the same general procedure as outlined in Smith

et al. (2003). The bottom panel of Figure 1 shows the results of the spectropolarime-

try averaged over a 2000 A-wide bin centered at 6000 A. The broad-band polarization

measurements were not corrected for the unpolarized starlight from the host galaxy

of Mrk 421 falling within the 3′′ × 10′′ spectral extraction aperture. Such a correc-

tion would increase the level of optical polarization, but does not affect the measured

polarization position angle. In addition to the spectropolarimetry, differential spec-

trophotometry was acquired with a slit width of 7.6′′. Again, a 10′′-wide extraction

aperture was used for both Mrk 421 and a comparison star calibrated by Villata et al.

(1998). No correction for the host galaxy was made to the reported R magnitudes

since the AGN still dominates the total flux observed in the larger aperture. The

largest flux variation observed in Mrk 421 during this period was ∼0.1 mag between

17 February (MJD 55244) and 19 February, 2010 (MJD 55246).

2.5. Radio Observations

Contemporaneous radio data were taken with the 40-m Owens Valley Radio Obser-

vatory (OVRO) telescope at 15 GHz, the 26-m University of Michigan Radio Astron-

omy Observatory (UMRAO) at 14 GHz and 8 GHz, and the 14-m Metsahovi Radio

Observatory at 37 GHz. Details of the observing strategy and data reduction are

given by Richards et al. (2011, OVRO); Aller et al. (1985, UMRAO); and Teraesranta

et al. (1998, Metsahovi). Mrk 421 is a point-like source for the three above-mentioned

single-dish radio instruments, which means that the measured fluxes are the flux den-

sities integrated over the full source extension. The light curves are shown in the

second from bottom panel of Figure 1 binned by individual observations.

3. RESULTS FROM THE EXCEPTIONAL FLARE ON MJD 55244

The flux state observed by VERITAS during the 17 February, 2010 (MJD 55244)

flare is extraordinary; it is the highest flux state in Mrk 421 ever observed in VHE

γ-rays. The peak fluxes measured above specific energy thresholds are given as ∼ 11

CU above 110 GeV, ∼ 15 CU above 200 GeV , ∼ 17 CU above 420 GeV, ∼ 21

CU above 600 GeV, ∼ 27 CU above 1 TeV; the higher flux with higher threshold

energy is due to Mrk 421 exhibiting a much harder spectrum during the flare than

the Crab Nebula. The “baseline” average flux from the MAGIC data just prior to the

main flare is 2.4× 10−10 photons cm−2 s−1 above 200 GeV, which is just below 1 CU.

15

The VERITAS γ-ray and RCT R-band optical data are the only two bands to have

high sampling rates during the night of this exceptional VHE flare. In this section,

we detail the results from the VERITAS observations along with the optical-VHE

correlation analysis.

3.1. Temporal Variability in the VHE γ-Ray Band

The high-statistics VERITAS data of the giant Mrk 421 flare enables construction

of finely-binned light curves. The energy threshold depends on the elevation of the

observations, increasing for smaller source elevation angles. 420 GeV represents the

lowest energy threshold common to the ∼5 hours of data taken during the night

of the flare (the full-night 2-minute-binned light curve with this threshold is shown

in Figure 4). For the first ∼2.33 hours of the night, the elevation of the source was

above 75◦, resulting in an energy threshold of 110 GeV for light curves generated with

these data. For this part of the flare night (∼140 minutes), we constructed 2-minute

and 5-minute-binned light curves above 110 GeV (shown in Figure 2) to characterize

any strong variability as discussed directly below. In addition, we constructed 2-

minute-binned light curves for three energy bands with equal statistics in each band:

a “low-energy” band, defined as 110 GeV < E < 255 GeV; a “medium-energy” band,

defined as 255 GeV < E < 600 GeV; and a ≥ 600 GeV “high-energy” band. We

investigate the fractional variability for these three bands in Section §4.2.

Table 1. Results from fits to the 2-minute light curves for the two burstsshown in Figure 2. The quoted (most likely) values represent the 50th per-centile, while the uncertainties are given as the 90th percentile of the posteriordistributions of the parameters. trise and tdecay are the doubling and halvingtimescales, respectively. The units for the normalization A are [10−9 photonscm−2 s−1 TeV−1]; κ is unitless.

Burst Fit A trise tdecay tpeak κ χ2/NDF

Function [min] [min] [min]

1 Exp 5.5+0.34−0.28 84+∞

−49 28+20−9.4 18+3.4

−5.2 – 18/12

GG 5.8+2.3−0.6 180+63

−100 55+91−23 17+2.6

−5.2 0.64+0.31−0.41 11/11

2 Exp 5.5+0.26−0.26 22+13

−6.7 65+13−9.6 44+2.3

−2.1 – 30/29

GG 6.6+1.6−0.88 30+30

−18 78+40−26 44+2.0

−1.3 0.47+0.28−0.19 27/28

Figure 1 shows the full set of VHE data during February, 2010 binned in nightly

bins. The first observations by VERITAS on 17 February, 2010 are likely to have

been taken after the onset of the flare; thus, we cannot make any statement about

the rise-time of the main flare. A decay function (N(t) = N0 ·2−t/tdecay , where tdecay is

the halving time) was fit to the four VERITAS data points in Figure 1 resulting in a

16

0 20 40 60 80 100 120 140Minutes (Since MJD 55244.311)

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5Fl

ux (>

110

GeV

) [10

9 cm

2 s1 ]

Burst 1Burst 22-minute bins5-minute bins

Figure 2. Light curve (2-minute and 5-minute bins) for VERITAS Mrk 421 data above 110GeV for the first 2.33 hours of observations on MJD 55244 where two “bursts” are identifiedvia a Bayesian Block analysis.

The dashed lines show an Exponential (Exp) function fit to the rise and fall of the twobursts using the 2-minute-binned light curve. The fit parameters, including the rise and

decay times are provided in Table 1.

halving timescale of ∼1 day. We fit the same function to the 10-minute binned data

resulting in a halving time of 1.17 ± 0.07 days, consistent with the nightly-binned

result.

A Bayesian Block analysis (Scargle et al. 2013) was applied to the > 110 GeV VER-

ITAS data from the flare night to look for any significant change points. Two peaks

or “micro-bursts” were identified in this manner in the first ∼140 minutes. Figure 2

shows a zoom in on this region with the peaks clearly visible in the first ∼95 minutes.

Burst 2 shows an apparent asymmetry which can be quantified via the method used

in Abdo et al. (2010); the symmetry parameter is found to be ξ = 0.50± 0.09 corre-

sponding to moderate asymmetry. We do not quote the asymmetry value for Burst

1 as we cannot be certain we observed the full rise of the burst. In addition, we fit

several functions to these data to determine the most likely rise and decay timescales

for the peaks. We test an Exponential (Exp), F(t) = A · exp[|t− tpeak|/trise,decay], and

the Generalized Gaussian (GG) burst profile from Norris et al. (1996) of the form,

F(t) = A ·exp[|t− tpeak|/trise,decay]κ. The most likely values and uncertainty of the pa-

rameters are determined using a Markov Chain Monte Carlo (MCMC) method with

the emcee tool (Foreman-Mackey et al. 2013a). The most likely values are taken as

the 50th percentiles, while the uncertainties are given as 90% confidence intervals of

17

3.5 4.0 4.5 5.0 5.5 6.0Flux (> 110 GeV) [ 10 9 cm 2 s 1 ]

2.0

1.9

1.8

1.7

1.6

1.5

1.4Sp

ectra

l Ind

ex

1

2

34

5

6

7

Burst 1

3.5 4.0 4.5 5.0 5.5 6.0Flux (> 110 GeV) [ 10 9 cm 2 s 1 ]

2.0

1.9

1.8

1.7

1.6

1.5

1.4

Spec

tral I

ndex

12

3

4

56

789

10

Burst 2

Figure 3. Photon index versus flux of the VERITAS detections of Mrk 421 over 5-minuteintervals shown separately for Burst 1 (left) and Burst 2 (right). The color shades representthe chronological progression of the bursts, with lighter colors corresponding to earlier times.The indices are obtained by a fit with an exponential cut-off power law (Ecut), where Ecutis fixed to the global value of 4 TeV.

the posterior distributions of the parameters. The fit results are provided in Table 1.

For Burst 2, the Generalized Gaussian profile with one more parameter than the Ex-

ponential function is not statistically preferred. In Section 5 we use the Burst 2 rise

time, trise = 22 minutes to place an upper bound on the effective size of the emis-

sion region as well as a lower bound on the Doppler factor when taking into account

compactness and opacity requirements of the emitting region.

3.2. Search for VHE hysteresis during flare

In order to investigate possible relationships between flux and photon index for the

> 110 GeV VERITAS data, coarser 5-minute bins were used for reducing statistical

uncertainties. Within each time bin, a flux estimation and a full spectral recon-

struction were performed. The Mrk 421 spectra are curved within each 5-minute bin,

therefore an exponential cutoff power-law function:

dN

dE= I0

(E

1 TeV

)Γ

exp

(− E

Ecut

), (1)

was used to reconstruct and fit the spectra, where Ecut is the cutoff energy. The

Ecut parameter was kept fixed to Ecut = 4 TeV, the value from a global fit. Figure 3

displays the resulting photon index versus flux representation of the VERITAS de-

tections of Mrk 421 for the two identified bursts. While there is some evidence for

a counter-clockwise hysteresis loop or a softer-when-brighter trend for Burst 1, the

photon index in Burst 2 is very stable while the flux rises and falls by a factor of

∼1.5.

18

The index-flux relationship for Burst 1 was assessed with simple χ2 tests using the

observed quantities against constant and linear models as the null hypotheses. A

constant model can be rejected with a p-value of 3.2× 10−5 (χ2/NDF = 30/6), while

the p-value for a linear model is 0.07 (χ2/NDF = 10/5). A χ2 difference test prefers

the linear model over the constant model with a p-value of 6.8 × 10−6. To test the

statistical robustness of this relationship, the observed data points were resampled

within the measurement uncertainties and the χ2 tests were repeated for 100,000

iterations. At a 90% confidence level, the constant model is rejected 99.7% of the time,

while the linear model is rejected 81.9% of the time. The p-value for the linear model

indicates the index-flux relationship for Burst 1 appears only marginally consistent

with a linear (softer-when-brighter) trend. The deviation from a linear trend could

be an indication of a more complicated relationship between the Mrk 421 index and

flux, such as a hysteresis loop.

These index-flux characteristics along with the asymmetry of Burst 2 can be used

to infer differing relationships between the cooling and acceleration timescales for the

bursts, which is further discussed in Section 5.

3.3. VHE γ-Ray and Optical Correlation Studies.

The RCT R-band optical data are the only dataset other than VERITAS to have

high statistical sampling during the night of the VHE highest state (MJD 55244).

Figure 4 shows the VERITAS 2-minute-binned data (blue) over the full energy range

(420 GeV < E < 30 TeV) commensurate with the low-elevation threshold, and

the R-band optical data (orange) overlaid. Visual inspection indicates an apparent

correlation between the two wavebands which warrants further investigation.

0 50 100 150 200 250 300Minutes (Since MJD 55244.311)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Flux

(> 4

20 G

eV) [

10

9 cm

2 s1 ]

VERITAS17.5

18.0

18.5

19.0

19.5

20.0

20.5 Flux Density (R-band) [ mJy ]

Optical

Figure 4. 2-minute-binned VERITAS > 420 GeV (blue) and RCT optical R-band (orange)light curves during MJD 55244.

19

We used the discrete cross-correlation function (DCCF) analysis following Edelson

et al. (1990) to test for time lags between the 2-minute-binned VERITAS and RCT

light curves. The DCCF was calculated after subtracting the mean from each light

curve and dividing the result by the standard deviation. There is a broad peak

apparent in the DCCF (turquoise points in Figure 5) centered at a lag time of roughly

45 minutes, with VHE γ-rays leading the optical.

In order to assess the statistical significance of features in the DCCF, including the

broad peak, Monte Carlo simulations were performed following the method by Em-

manoulopoulos et al. (2013). First, as described in Appendix A, the Power Spectral

Density (PSD) was constructed and fitted for both the VERITAS and optical light

curves. Next, the best-fit VERITAS PSD (P (f) ∝ f−1.75) was used to generate

100,000 random light curves. The random light curves were then paired with the

observed optical light curve to calculate a DCCF for each iteration.

-100 -50 0 50 100Lag [ minutes ]

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Corre

latio

n Co

effic

ient

1 2 3

0.00

0.01

0.02

0.03

0.04

0.05

0.06

DCF

Frac

tion

Figure 5. Simulated DCCFs for VERITAS (> 420 GeV) and optical R-band light curves.The DCCF from observations is in turquoise. “DCF fraction” represents the fraction oftimes a simulated DCCF falls in a given lag-time and correlation-coefficient bin (shownwith the 2D histogram color map). The DCF fraction histogram (representing a PDF) isintegrated to obtain the confidence levels. The black, blue, and red dashed lines show the1σ, 2σ, and 3σ levels respectively. A positive lag time corresponds to a delay in the opticallight curve with respect to the VERITAS light curve.

Figure 5 shows the resulting simulated DCCFs binned into a 2D histogram of cor-

relation coefficient versus lag time bins. The bin contents of the 2D histogram are

normalized such that for a fixed lag time, each correlation coefficient bin gives the

fraction of all DCCFs falling within the bin, and the bin contents along the vertical

axis will sum to one. Significance levels are estimated by integrating the probability

density function (PDF) represented by the 2D histogram of simulated DCCFs. The

20

VERITAS-optical DCCF shows evidence for a signal at lag times of 25–55 minutes.

The significance of the correlation is ∼ 3σ. The use of an observed light curve (in

this case, the optical R-band) in the significance level estimation is a conservative ap-

proach. If simulated light curves are generated from the optical PSD (P (f) ∝ f−1.85),

the correlation significance increases to ∼ 4σ. We note, however, that the PSD fit

errors are large, hindering a good characterization of the uncertainties on the signifi-

cance of the correlation.

3.4. Autocorrelation Analysis with the VHE Flare

The VHE flux from Mrk 421 shows clear intra-night variability during the night of

the flare on 17 February, 2010 and the PSD analysis for VERITAS from Appendix A

shows a power spectrum of pink noise (or flicker noise) with P (f) ∝ f−1.75. However,

these results are limited to shortest timescales of ∼ 500 seconds.

A modified auto-correlation function (MACF) proposed in Li (2001) and extended in

Li et al. (2004) could provide improved sensitivity to short variations of the VHE flux.

Details of the method can be found in Appendix B. Though Bolmont et al. (2009)

used the method to search for signatures of potential Lorentz invariance violation

(LIV) in the 2006 PKS 2155-304 flare, it is a novel technique for VHE variability

studies. In Appendix B, we have applied the MACF to the night of the VHE flare

(Epoch 3; see Section 4.1) using all events above an energy threshold E = 420 GeV.

No critical timescale is observed on these short timescales, but the data are consistent

with a stochastic process or “pink noise” corroborating the VERITAS PSD results

found at longer timescales in Appendix A. Probed by the combination of these two

techniques, this is the first time that this stochastic behavior has been shown to exist

in a blazar on the full range of timescales from seconds to hours.

4. RESULTS FROM FULL FEBRUARY 2010 MULTIWAVELENGTH DATA SET

4.1. Light Curves

In this section, we focus on the multiwavelength light curves of Mrk 421 for Febru-

ary, 2010. Figure 1 shows the light curves for each waveband participating in the

campaign. The VHE data are shown averaged over the full set of observations for

a given night spanning durations between ∼twenty minutes and ∼six hours; Fermi -

LAT and MAXI data are shown with daily binning; all other light curves are binned

by individual exposures. To study the flux properties of the VHE data in more de-

tail, the entire combined MAGIC and VERITAS dataset from Figure 1 was split into

multiple epochs. MAGIC data are available for several days leading up to the flare.

These Epoch 1 data (MJD 55232-55240) are used as “baseline” VHE data to which

we compare the flaring period and its decline. Epoch 2 (MJD 55240-55243) has no

VHE data, however it is used to study the behavior of the X-ray and HE data as the

flare builds up in these bands (see Section 4.4). Epoch 3 comprises the main flare

(MJD 55244) showing extraordinary overlap between the VHE and optical data en-

abling the correlation analysis shown in Section 3.3. Epochs 3-7 (MJD 55244, 55245,

21

17 February 18 February 19 February 20 February 21 February 22 February

100

150

200(3

-15

keV

)[ s

1 ]

RXTEVERITASMatched

0.0

0.5

1.0

1.5

(> 3

50 G

eV)

[ 10

9 cm

2 s

1 ]

Epoch 3 Epoch 4 Epoch 5 Epoch 6 Epoch 7

0.05

0.10

0.15

0.20

Har

dnes

s R

atio

55244 55245 55246 55247 55248 55249Date [ MJD ]

3.0

2.5

2.0

1.5

Inde

x

Figure 6. Detailed 10-minute-binned light curves for VHE (VERITAS - blue) and X-ray(RXTE - brown) data for Epochs 3-7: the VHE flare (Epoch 3) followed by the three VHEdecline epochs (Epochs 4-6) and a final epoch during which RXTE count rates are elevatedagain. Matched - red points distinguish data where there is strictly simultaneous overlapbetween RXTE and VERITAS observations. The top panel shows as a function of time, theRXTE count rate and VERITAS flux light curves, while the bottom panel shows the RXTEhardness ratio between the 5-15 keV and 3-5 keV bands and the VERITAS photon indexfrom power-law fits between 350 GeV and 3 TeV. We note that there are no simultaneousX-ray data during the VHE flare (Epoch 3) and there are no VHE data during Epoch 7.Regions of overlap are indicated by gray hatches and their behavior studied in Section 4.3.Color-shaded regions are used for a more in-depth X-ray hardness ratio-count rate studyillustrated by the bottom panels of Figure 13.

55246, 55247, 55248) are shown in Figure 6 which displays 10-minute-binned light

curves for both VHE and RXTE X-ray data (top two panels). Epochs 3-6 comprise

only VERITAS data in the VHE band (shown above 350 GeV as the lowest com-

mon threshold) and are, respectively, during the VHE flare and just afterwards in

three decline epochs. Epochs 4-7 comprise RXTE data in the 3-15 keV band where

Epochs 3-6 overlap with the VERITAS data during the decline phases and a subse-

quent rise in RXTE data is seen in Epoch 7; no VHE data are available in this last

epoch. During periods where strictly simultaneous data were obtained, we matched

the start and stop times of each time bin between the VERITAS and RXTE light

curves. These VHE and X-ray light curves, along with the VERITAS photon indices

and RXTE hardness ratios shown in the bottom two panels of Figure 6 are used in

22

more detailed studies in Section 4.3. However, first we compare variability properties

across all participating wavebands shown in Figure 1.

2.5 0.0 2.5 5.0 7.5 10.0 12.5log E [eV]

0.0

0.2

0.4

0.6

0.8

F var

RadioOpticalSwift-XRTRXTEMAXIFermi-LATVHE

2.5 0.0 2.5 5.0 7.5 10.0 12.5log E [eV]

0.0

0.2

0.4

0.6

0.8

F var

(MJD 55244) Optical(MJD 55244) VERITAS 110-255 GeV(MJD 55244) VERITAS 255-600 GeV(MJD 55244) VERITAS >600 GeV

Figure 7. Left: Fractional variability for each waveband over the full dataset shownin Figure 1 (key in top left). The VHE band uses the nightly averaged binning fromFigure 1 for both the VERITAS and the MAGIC light curves; the optical and radio bandsinclude observations from all participating observatories displayed in Figure 1. Right: Fvarcalculated for the 2-minute-binned light curves produced for the optical and three separateVERITAS energy bands on the night of the main flare as shown in Figure 4 (key in topleft).

4.2. Multiwavelength Variability

We calculated the fractional root mean square variability amplitude, Fvar (Edelson

et al. 1990; Rodrıguez-Pascual et al. 1997) – as defined by Equation 10 in Vaughan

et al. (2003) with its uncertainty given by Equation 7 in Poutanen et al. (2008) – for

each available band, with the results shown in Figure 7. The Fvar calculation was

performed for the full duration of the light curves shown in Figure 1 and, separately,

for the 2-minute-binned optical and VERITAS (3 bands) light curves from the night

of the giant flare (MJD 55244). Note that the four radio bands are shown under a

single point covering the energy range of the bands, as no excess variance (Fvar = 0)

was found in any of the bands.

The Fvar values from the full light curves spanning the month of February, 2010

increase from radio to optical to X-ray, drop again for the HE band, and then show

maximal Fvar for the VHE band. This “double-humped” Fvar characterization, which

has been observed in Mrk 421 during low and high activity (Aleksic et al. 2015a,b;

Balokovic et al. 2016), could reflect the global difference in cooling time between the

populations of electrons underlying the different bands. However, no strong conclu-

sions can be drawn from these values as the integration times differ drastically for

the light curves from different instruments potentially introducing large biases.

23

The Fvar values for the optical and VERITAS light curves from MJD 55244 are

more reliable for inter-band comparison, showing higher values for the VERITAS

bands compared to the optical and an indication of an increasing trend with energy

within the three VERITAS bands (though the p-value of a χ2 difference test between

a linear and constant fit is 0.13). If the particle-cooling timescale (with inverse-

Compton scattering or SSC) is longer than the dynamical timescale of the emission

region, the increasing Fvar with energy observed in the VHE can be related to the

difference in cooling times between particles of different energies. The higher-energy

particles will cool faster, producing larger variability and a correspondingly higher

Fvar value for a given timescale than lower-energy particles that cool more slowly.

There is a large contrast between the impressive flux variations at high energies

and the muted behavior both in optical flux and linear polarization seen in Figure 1.

The optical data show a smooth decrease of 20% over the entire period. Two “fast”

variations (1-2 day timescales) of about 15-20% are noted: one on MJD 55236 (8

February, 2010) in Epoch 1 in the “preflare” time interval and the other in Epoch 2

on MJD 55244 (16 February, 2010), the night before the ∼11 CU (above 110 GeV)

flare measured with VERITAS. This latter fast optical variation is during the period

where the HE and X-ray observations show some evidence for correlation (see Section

4.4). It is interesting to note that, while the source clearly stayed high on 17 February,

2010 in X-rays and VHE, the optical flux diminished to values just slightly higher

than the pre/post flare flux.

The optical polarization for Mrk 421 increased from P=1.7% to 3.5% during the

VHE flare. No change in polarization position angle was detected over the same

period, although larger (∼ 20◦) position angle swings are observed just prior to and

after the VHE flare. In general, both the variability in optical flux and polarization

are mild during this period, with P=1–3.5% and θ = 125◦−155◦. For comparison, the

Steward Observatory monitoring data for Mrk 421 obtained during the January, 2010

and March, 2010 observing campaigns show the blazar to be more highly polarized.

For 14-17 January, 2010, P=3.7-5.0%; θ = 157◦−163◦ and during 15-21 March, 2010,

P=3.1-4.9% with θ = 114◦−130◦. In addition, the object was about 0.3 mag brighter

during the January, 2010 campaign compared to the February measurements, while

it was < 0.1 mag fainter in March, 2010.

There are no signs of unusual activity in the radio observations of Mrk 421 with the

instruments that participated in this campaign (UMRAO, Metsahovi and OVRO)

over the two weeks before and the two weeks after the main VHE flare. However,

no observations were taken during the VHE flare night. High-resolution Very Long

Baseline Array (VLBA) observations of Mrk 421 were collected on 11 February, 2010

as part of the Monitoring Of Jets in Active galactic nuclei with VLBA Experiments

(MOJAVE) program (Lister et al. 2018). MOJAVE data on Mrk 421 are also available

from 17 December, 2009 and 12 July, 2010 observations. The 15 GHz MOJAVE im-

ages show significant extended structures associated with the source. The emergence

24

of a potentially new component within the Mrk 421 mas radio jet over the month

following the giant flare was reported by Niinuma et al. (2012) using the Japanese

Very Long Baseline Interferometry (VLBI) Network, and also by Jorstad et al. (2017)

using observations from the VLBA Blazar program from Boston University. However,

we cannot conclude that any of the components from MOJAVE or the VLBI observa-

tions are associated with the 17 February, 2010 VHE flare. The relative VLBA flux

density, SVLBA/Stotal (Stotal is the filled-aperture single-antenna flux density) from the

11 February, 2010 MOJAVE and 12 February, 2010 OVRO observations is comparable

to the average historical value of ∼ 0.75 from Kovalev et al. (2005). The parsec-scale

jet direction reported in Jorstad et al. (2017) is about −25◦ and the polarization

angle of the radio knot B1 is about −35◦, both angles being approximately the same

to those reported in Figure 1 for the optical EVPA taking into account the ambiguity

of EVPA with respect to π.

4.3. VHE γ-Ray and X-ray Correlation Studies.

By visual inspection of Figure 1, we cannot ascertain whether the VHE flare was

observed at its peak or on the decline. Furthermore, there is no overlapping Swift-

XRT or RXTE data during the night of the highest VHE flux. Unfortunately we

therefore cannot determine any correlation between X-ray and VHE at the peak

observed in either band. There are only three Swift-XRT exposures over the first two

days of the decline, averaging 3.6 ks per exposure. Kapanadze et al. (2018) analyzed

these data along with all available Swift-XRT data for the period 2009-2012. However,

RXTE data comprise eight short (average 3.6 ks) and five long (from 9.8 ks to 48.2

ks) observations during the decline Epochs 4-6 which overlap with VERITAS data.

We thus use the RXTE data for our in-depth VHE-X-ray studies; both these data sets

are shown in Figure 6 with a zoomed-in version overplotting RXTE and VERITAS

data for Epochs 4-6 in the top three panels respectively of Figure 8. Clear inter-

day variability is evident in both the VHE and X-ray bands with Epoch 4 mainly

comprising a strong decay in an X-ray flare (40% drop in PCA rate) while the VHE

shows a slight rising trend, Epoch 5 catches the tail of another smaller X-ray flare

followed by a rise - both mirrored in the VHE, and Epoch 6 shows minimal X-ray

and VHE variability.

In Figure 6 we also show the VERITAS photon indices as well as the RXTE hardness

ratio in 10-minute time bins (lower two panels). The hardness ratio (HR) is taken

between the 5-15 keV and 3-5 keV bands while the VHE indices are found from a

power-law fit between 350 GeV and 3 TeV. Here we note that the overall trend for the

X-ray data is an increase in the hardness ratio while the source in general is becoming

steadily weaker in the X-rays (top panel of Figure 6). On the other hand, the VHE

data show no general trend during the decline phases of the flaring period. However,

there are periods when the VHE indices become significantly harder than Γ ∼ −2,

indicating the VHE emission is in part below the IC peak frequency. Some of these

25

55245.0 55245.2 55245.4 55245.6 55245.8 55246.0

0.25

0.50

0.75(>

350

GeV

) [

109 c

m2 s

1 ] Epoch 4VERITASRXTE

55246.0 55246.2 55246.4 55246.6 55246.8 55247.0

0.25

0.50

0.75

(> 3

50 G

eV)

[ 10

9 cm

2 s1 ] Epoch 5

55247.0 55247.2 55247.4 55247.6 55247.8 55248.0MJD [ days ]

0.25

0.50

0.75

(> 3

50 G

eV)

[ 10

9 cm

2 s1 ] Epoch 6

100

125

150 (3-15 keV)[ s

1 ]

100

125

150 (3-15 keV)[ s

1 ]

100

125

150 (3-15 keV)[ s

1 ]

0.02 0.04 0.06 0.08 0.10 0.12 0.14RXTE Hardness ratio

2.6

2.4

2.2

2.0

1.8

1.6

VE

RIT

AS

Inde

x

Epoch 4Epoch 5Epoch 6

85 90 95 100 105 110 115 120 125Count Rate (3-15 keV) [ counts s 1 ]

0.1

0.2

0.3

0.4

0.5

0.60.7

Flux

(> 3

50 G

eV) [

10

9 cm

2 s

1 ]

= 3.0

= 2.0

= 1.0

Epoch 4Epoch 5Epoch 6

Figure 8. Top three panels: The detailed 10-minute-binned light curves for Epochs 4-6from Figure 13 with the VHE (VERITAS-blue) and the X-ray (RXTE -brown) overplottedin the same panel to better highlight trends described in the text. From top to bottom, thepanels are Epoch 4, Epoch 5 and Epoch 6. Bottom two panels: VERITAS photon indicesversus RXTE hardness ratios (left) and VERITAS and RXTE flux-flux correlation (right)plots based on the 10-minute-binned light curves for each of the epochs in the above panels.Only points are plotted here that correspond to the “matched” points in Figure 6 wherethere is strictly simultaneous overlap between RXTE and VERITAS observations. Graylines and color gradients are intended to guide the chronological progression of the points.The hardness ratio is taken between the 5-15 keV and 3-5 keV bands while the VHE indicesare found from a power-law fit between 350 GeV and 3 TeV.

exceptionally hard indices correspond to instances in which the VHE flux is at its

weakest in this dataset. This is especially true towards the end of Epoch 6.

The bottom left panel of Figure 8 looks at the VHE index versus X-ray hardness ratio

over the full decline phase but restricted to data pairs where there is an exact time

match between the VERITAS and RXTE data (gray bands in Figure 6). The data

26

suggest a clustering around distinct states that represent “snapshots” of the evolving

system over several days. The cluster of extremely hard VHE indices and high X-ray

HR values corresponds to a weak-flux state observed in both bands. Though unusual,

these observations could indicate both the synchrotron and IC peaks shifting together

to higher frequencies (without an increase in peak luminosity).

As there are substantially more X-ray data than VHE data throughout the decline

epochs, in Appendix C we carry out a detailed examination of the X-ray data search-

ing for evidence of hysteresis in the relationship between HR and the count rate. All

epochs show a considerably different evolution of the hardness ratio with flux, with

a variety of loops and trends exhibited even as the overall increase in HR is seen as

the source weakens in X-ray across the decline (as noted in Figure 6). The standard

harder-when-brighter scenario is only distinctly observed in Epoch 5.

Table 2. Results from fits to the data for the left panel of Figure 8with the relation Fγ ∝ FΓ

X , where Fγ is the VERITAS flux above 350GeV in units of 10−9 cm−2 s−1, FX is the RXTE count rate between3 keV and 15 keV, and Γ is the index. The Pearson’s ρ is shown alongwith the p-value for each fit.

Dataset Γ χ2/NDF ρ p-value

Full 3.3± 0.2 220/30 0.76 3.6× 10−7

Epoch 4 (all) 1.5± 0.07 16/8 0.22 0.52

Epoch 4 (first 4) −1.6± 0.18 0.85/2 -0.86 0.14

Epoch 4 (first 5) −3.2± 1.2 5.4/3 -0.78 0.17

Epoch 4 (last 6) 0.7± 0.09 0.34/4 0.57 0.24

Epoch 5 (all) 2.0± 1.0 37/7 0.35 0.36

Epoch 5 (first 4) 2.5± 0.8 1.5/2 0.92 0.078

Epoch 5 (last 5) 1.6± 1.0 1.3/3 0.74 0.15

Epoch 6 (all) 1.9± 0.1 6.9/11 0.48 0.095

Epoch 6 (no last point) 1.6± 0.3 2.1/10 0.64 0.024

To further investigate the flux-flux relationship between the synchrotron and IC

peaks during Epochs 4-6, we show the VHE–X-ray flux-flux plot in the bottom right

panel of Figure 8 for each epoch, where the X-ray and VHE data are simultaneous

(indicated by the gray bands in Figure 6). We also show the linear, quadratic and

cubic slopes corresponding to the relation Fγ ∝ F ΓX with fit values shown for each

epoch displayed in Table 2, along with the slopes for subsamples of the data in each

epoch as well as the full dataset. For simple SSC behavior, we would expect to

see correlation between the X-ray and VHE emission with a linear correlation slope

indicating the system was in the Klein-Nishina (KN) regime (Tavecchio et al. 1998).

27

In fact, the VHE-X-ray flux-flux plot shows inconsistent behavior across the three

epochs. When considering the first four points, Epoch 4 shows a hint of an anti-

correlation between the VHE and X-ray bands which would be very inconsistent with

a single-zone SSC model. Taking the last six points of Epoch 4, no correlation is seen

– the VHE stays roughly constant in flux as the X-ray dims. Epoch 5 captures a

fast decrease in both VHE and X-ray, followed by a less dramatic rise in both bands.

Both the fall and rise states show a correlation between the two bands, however with a

∼quadratic behavior in both “cooling” and “acceleration”. Epoch 6 shows an erratic,

uncorrelated relationship in time between the X-ray and VHE bands, though with a

global fit nearly quadratic in slope. Taken together, the range of behavior across the

decline epochs between and within the X-ray and VHE bands is difficult to interpret

as the evolution of the system in the context of a single-zone SSC model.

4.4. HE γ-Ray and X-ray Correlation Studies.

By inspection of the light curve in Figure 1, Epoch 2 shows an increase in both

the MAXI X-ray and Fermi -LAT HE γ-ray daily binned fluxes the day prior to the

VHE flare observed with VERITAS. A simple test for variability was performed on

the Fermi -LAT light curve. This yielded an improvement in log-likelihood over a

constant model equivalent to χ2 = 39.2 for 23 degrees of freedom, corresponding to

a p-value = 0.018. The MAXI light curve is clearly variable (χ2/NDF = 930/23;

p-value∼0).

A preliminary cross-correlation analysis using Fermi -LAT “Pass7” P7SOURCE V6

event selection and instrument response functions found that the lag between the X-

ray and HE γ-ray light curves was consistent with 0 days (Madejski et al. 2012). We

performed an analysis using the “Pass8” Fermi -LAT data and IRFs corresponding to

those used to generate the Fermi -LAT light curve in Figure 1. A linear correlation

coefficient was calculated for the time-matched MAXI and Fermi -LAT fluxes, result-

ing in a mean value of ρ = 0.54 ± 0.12. The mean value and the 1σ uncertainties

of the linear correlation coefficient were determined by resampling both light curves

within measurement uncertainties over 100,000 iterations.

To further investigate this potential correlation, we conducted a DCCF analysis

between MAXI–Fermi -LAT light curves in the manner described in Section 3.3. In

this case, the PSD from the MAXI light curve was fit using the method by Max-

Moerbeck et al. (2014) and the best-fit MAXI PSD (P (f) ∝ f−1.95) was used to

generate 100,000 random light curves paired with observed Fermi -LAT light curve

(the conservative approach), with the results shown in Figure 9.

We find a ∼ 2σ correlation at a lag of ∼ 0 days. The confidence level of the

correlation at ∼ 0 days is considerably higher (∼ 4σ) if light curves are simulated

from PSDs for Fermi -LAT as well (with best-fit PSD, P (f) ∝ f−1.75). The PSD fit

errors are very large, however, making it difficult to characterize the uncertainties on

the significance of the correlation.

28

8 6 4 2 0 2 4 6 8Lag [ days ]

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Corre

latio

n Co

effic

ient

1 2 3

0.00

0.01

0.02

0.03

0.04

0.05

0.06

DCF

Frac

tion

Figure 9. The DCCF calculated from the observed MAXI–Fermi -LAT light curves isshown with turquoise points. Correlation significance levels (shown with dashed lines) areestimated through a Monte Carlo method. During each iteration, the observed Fermi -LAT light curve is paired with a light curve simulated from the MAXI PSD to calculate asimulated DCCF. “DCF fraction” represents the fraction of times a simulated DCCF fallsin a given lag-time and correlation-coefficient bin (shown with the 2D histogram color map).The DCF fraction histogram (representing a PDF) is integrated to obtain the confidencelevels.

5. DISCUSSION AND CONCLUSIONS

The VHE flare observed from Mrk 421 in February, 2010 is an historically significant

flare. During the night of the giant flare observed with VERITAS, Mrk 421 reached a

peak flux of about 27 CU Units above 1 TeV. This episode rivals the brightest flares

observed from any source in VHE γ-rays, including the extraordinary flare of PKS

2155-204 in 2006 detected by H.E.S.S. (Aharonian et al. 2009) and the 27 February,

2001 flare of Mrk 421 seen with Whipple (Krennrich et al. 2001; Acciari et al. 2014).

Another exceptionally strong flare in Mrk 421 was detected by both VERITAS and

MAGIC in April, 2013 (Cortina & Holder 2013). As extreme as the currently reported

flare is, it is unclear from the analyses described in this paper and summarized below

whether this represents a fundamentally different behavior state for this object or

just an extreme end of the same underlying processes that have yielded the range of

behavior previously reported.

5.1. VHE-Optical Band Correlation

A cross-correlation analysis was performed between the VHE and optical bands dur-

ing the night of the VHE flare. The observed optical and VERITAS 2-minute-binned

light curves exhibit a 3σ − 4σ significance correlation with an optical lag of 25-55

minutes centered at 45 minutes. Such behavior can be accommodated under a single-

29

zone SSC scenario, in which the emission in both VHE and optical bands is produced

by a single distribution of electrons. Under this scenario, the optical lag could be ex-

plained by the slower cooling of the less energetic electrons that underlie the optical

data compared to the electrons responsible for the VHE emission (Boettcher 2019).

The lag timescales can be used to set an additional constraint on the magnetic field

strength for future SED modeling efforts.

VHE-optical correlation has been previously observed in HBLs, but not with a lag

and not at the short timescales probed by this unprecedented dataset. For example,

the 2008 multiwavelength campaign on PKS 2155-304 reports a > 3σ correlation

between the H.E.S.S. data and the V , B, and R optical bands on daily timescales

and with no lag (Aharonian et al. 2009). It is more common to observe a correlation

between the HE and optical which is likely explained by both bands arising from the

same electron population in the simple SSC model (Cohen et al. 2014).

5.2. Fast Variability in VHE γ-rays

The exceptional brightness of the flare on 17 February, 2010 in VHE enabled VERI-

TAS to produce 2-minute-binned light curves with 10σ significance in each bin yield-

ing strongly-detected, short-term variability. The variable emission within the first

95 minutes of VERITAS observations on that night can be described by at least two

successive bursts. Burst 2 is characterized by an asymmetric profile with a faster rise

time, followed by a slower decay. This behavior has been previously observed (e.g.,

Zhu et al. 2018) and is typically attributed to emission from electrons with longer

cooling than the dynamical timescales, assuming both the cooling and dynamical

timescales are much longer than the acceleration timescale. Under this scenario, the

flare rise time is related to the size of the emission region (e.g., Zhang et al. 2002).

Assuming the above conditions, we used the rise timescale of Burst 2 to place an

upper limit on the size of the emission region associated with the burst, RB:

RB ≤ctvarδ

1 + z, (2)

where c is the speed of light, tvar is the variability timescale, and δ is the Doppler

factor. Using the most likely Burst 2 rise time of 22 minutes for tvar, we obtained

δ−1RB . 3.8× 1013 cm.

Furthermore, the time variability of the VHE flux, in conjunction with compactness

and opacity requirements of the emitting region, can be used to give an estimate of

the minimum Doppler factor of the ejected plasma in the jet of the blazar. Following

Tavecchio et al. (1998) and Dondi & Ghisellini (1995), the minimum Doppler factor

was calculated using:

δmin >

[σT

5hc2dL(1 + z)2βF (ν0)

tvar

]1/(4+2β)

, (3)

30

where σT is the Thomson scattering cross-section, dL is the luminosity distance of

the source, z is the redshift, tvar is the observed variability timescale, and F (ν0) and

β are the flux and spectral index, respectively, of the target photons of the γ-rays for

pair production.

To estimate the Doppler factor limit, we used the following parameters: the ob-

served variability timescale tvar,VHE = 22 minutes; the γ-ray photon energy Eγ =

110 GeV, corresponding to a target photon frequency of 6.0 × 1014 Hz (500 nm)

for maximum pair-production cross section; and the spectral index, β = −0.16,

and F (ν0) = 1.35 mJy of the low-energy photons derived from the 3 Swift-UVOT-

band observations during MJD 55244-55246. The latter value was obtained using,

F (ν0) = Fuvw1(νuvw1/ν0)β. Assuming these parameters, the derived Doppler factor

lower bound is δmin & 33. The fast variability measured with this dataset results in a

larger Doppler factor compared to B lazejowski et al. (2005), where a lower limit on

the Doppler factor of δmin & 10 was obtained with an ∼hour-scale time variability in

the VHE data from the 3-CU flare of Mrk 421 during April, 2004.

For the overall system to be consistent with reported lower Doppler factors from

VLBI measurements, results from fast flares such as that reported here indicate that

the γ-ray emission zone may be smaller than the jet-cross section. For example,

Giannios (2013) suggests that rapid∼minute-scale flares on an “envelope” of day-scale

flares can be due to large plasmoids created during a magnetic reconnection event.

However, Morris et al. (2018) show that while such a “merging plasmoid” model can

explain the VHE light curve from the 2016 fast flare from BL Lac (Abeysekara et al.

2018), it has difficulty reproducing the SED.

A potential counter-clockwise loop (known as spectral hysteresis), or a harder-when-

weaker trend, is present in the index versus flux representation for Burst 1, while the

photon index is essentially constant for Burst 2 even as the flux changes by a factor

of ∼1.5. Spectral hysteresis can occur as a result of competing acceleration, cooling,

and dynamical timescales, which determine how the effects of particle injection into

an emitting region translate to the observed photons (Kirk et al. 1998; Li & Kusunose

2000; Bottcher & Chiang 2002). Counter-clockwise hysteresis is related to a case in

which dynamical, acceleration, and cooling timescales are comparable. The change

in the number of emitting particles in this scenario is determined by the acceleration

process, which proceeds from lower to higher energies and leads to higher-energy

photons lagging behind the lower-energy photons.

A modified auto-correlation analysis is applied to the VERITAS data on the night of

the flare to look for potential variability on short timescales; however, no significant

time structures are found within 10–60 s timescales. Combining this result with

timescales probed by the VHE PSD analysis, we conclude that the VHE emission is

consistent with a pink noise characterization over a wide range of timescales – from

∼seconds to ∼hours. Power-law PSDs in blazars have been detected in X-ray as well

as VHE and are indicative of an underlying stochastic process (Aharonian et al. 2007).

31

A power-law PSD could also point to a self-organizing criticality (SOC) system, such

as magnetic reconnection, as the underlying physical process responsible for the flaring

behavior observed for Mrk 421 (Lu & Hamilton 1991; Aschwanden 2011; Kastendieck

et al. 2011). A recent study of Mrk 421 flares extracted from archival XMM-Newton

X-ray data spanning from 2000 to 2017 is consistent with the expectations for a SOC

model, thus lending support to the magnetic reconnection process driving blazar

flares (Yan et al. 2018). Additionally, the flatness of the PSD indicates that the turn-

on/turn-off timescale of mini-flares can be below an hour and generally has a wide

probability distribution extending from sub-hour timescales to entire nights (Chen

et al. 2016).

5.3. Multiwavelength Correlation Studies

In addition to the optical-VHE correlation study, several other intra-band and multi-

band correlation studies were carried out. The decay of the flare in the VHE and

X-ray bands occurs over the course of four days. Correlation studies between VHE

(VERITAS) and X-ray (RXTE ) bands show a diverse and inconsistent range of be-

havior across the decline epochs. The flux-flux relationship between the synchrotron

peak (as probed by the X-ray data) and the IC peak (as probed by the VHE data)

moves in Epoch 4 from an indication of anti-correlation to no correlation. B lazejowski

et al. (2005) report a lack of correlation seen in day-scale coincident VHE (Whipple)

and X-ray (RXTE ) data, which is potentially explained by an X-ray flare leading

the VHE flare by 1.5 days. The dataset reported in our work indicates lack of cor-

relation between X-ray and VHE on the ∼10-minute timescales probing potentially

quite different mechanisms. To our knowledge, an anti-correlation between the X-ray

and VHE has never before been reported for Mrk 421. Epoch 5 shows a ∼quadratic

behavior in Fγ ∝ F ΓX most notably in the “cooling” segment of the epoch. This be-

havior has been seen before in both Mrk 421 (Fossati et al. 2008) as well as in the

exceptional flare in PKS 2155-304 (Aharonian et al. 2009) and is not consistent with

the linear relationship expected from a system scattering in the Klein-Nishina regime

(Aharonian et al. 2009). However, Thomson scattering into VHE photon energies

requires unacceptably large Doppler factors (Katarzynski et al. 2005).

The RXTE results indicate spectral hardening as the source becomes fainter over

this period. Such behavior can be an indication of the synchrotron peak shifting

to higher frequencies as the flare decays, which would be unusual, or the possibility

that the synchrotron photons in the keV band soften first, uncovering a population

of harder photons produced in the keV band by the IC process at the beginning of

the flare (Li & Kusunose 2000). On the other hand, no clear long-term trends are

apparent in the VHE photon index as the flare decays. Nonetheless, it is interesting to

note that the VHE indices become harder than −2 at times during the decay period,

indicating the Compton peak moves into the TeV regime even as the overall VHE

flux is decreasing. The fact that both the X-ray and VHE data show a harder-when-

32

weaker trend at the same time may indicate that both peaks have shifted and the

source has temporarily become an extreme HBL (Costamante et al. 2001; Bonnoli

et al. 2015; Cerruti et al. 2015). Time-dependent extreme HBL behavior has recently

been reported for Mrk 501 (Ahnen et al. 2018) though changing on yearly timescales.

A correlation between HE and X-ray (MAXI) bands was observed on daily

timescales. We found a ∼ 2σ correlation at a lag of ∼ 0 days while a less con-

servative approach yielded ∼ 4σ. While unusual, HE and X-ray correlations have

been seen in other jetted systems including NGC 1275 and can indicate, for example,

a fresh injection of electrons into the emission region (Fukazawa et al. 2018).

5.4. Multiwavelength Variability

A study of the energy dependence of the fractional variability (Fvar) across all

participating instruments resulted in a “double-humped” structure that seems to be

characteristic for Mrk 421 in both flaring and quiescent states (Aleksic et al. 2015a,b;

Balokovic et al. 2016). However, this is quite different from the Fvar characterization

seen in the other well-studied nearby HBL, Mrk 501, where a general increase in

variability as a function of energy has been observed (Aleksic et al. 2015c; Ahnen

et al. 2017b, 2018). While a strict comparison is difficult due to the vastly different

integration times for the participating instruments in each campaign, the different Fvardependence on energy between the two sources is likely attributed to the difference

in the Fvar values in the X-ray band, with lower X-ray Fvar values typically seen in

Mrk 501. This could indicate that the X-ray instruments more often probe the rising

edge of the synchrotron peak for Mrk 501 than for Mrk 421 which would be consistent

with the synchrotron peak excursions to more extreme HBL regimes seen in Mrk 501

(Nieppola et al. 2006; Ahnen et al. 2018). The upcoming work studying the SEDs

constructed from these data can further elucidate these observations.

Software: VEGAS (Cogan 2008), Event Display (Daniel 2008), MARS (Moralejo

et al. 2009), Fermi Science Tools5, HEASoft (FTOOLS+XANADU) (HEASARC

2014), REX6, Xspec (Arnaud 1996), emcee (Foreman-Mackey et al. 2013b)

5 http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/6 http://heasarc.gsfc.nasa.gov/docs/xte/recipes/rex.html)

33

APPENDIX

A. POWER SPECTRAL DENSITIES

Figure 10. 2-minute-binned VERITAS > 420 GeV (top) and optical R-band (bottom)light curves. Left panels are the observed light curves while Right panels show the lightcurves after long-term trend removal.

For the Mrk 421 VERITAS and optical light curves, Power Spectral Densities (PSDs)

were calculated using the FFT method available through the POWSPEC program

within the XANADU X-ray astronomical spectral, timing, and imaging software

package.7 PSDs were calculated with both the observed VERITAS and optical light

curves as well as for those where the long-term trends have been modeled and removed.

Trend removal was done to avoid potential contamination of higher-frequency signal

by lower frequencies. A piecewise continuous linear function – represented by a linear

spline with a single node at the best-fit location – was used to model and subtract

the long-term trend in each light curve. The observed and detrended light curves are

presented in Figure 10. The entirety of both the VERITAS and optical light curves

was used. Light curves are split into intervals within which the power spectra were

independently calculated and later averaged.

The uncertainties on power in individual frequency bins were calculated as the

standard deviation of the average of the power from different intervals. The resulting

VERITAS and optical PSDs with and without detrending are displayed in Figure 11.

The best-fit power-law spectral indices for the VERITAS and optical PSDs were

estimated with the method by Max-Moerbeck et al. (2014) resulting in values of

−1.75 and −1.85, respectively. The detrended light curves were used to determine

7 https://heasarc.gsfc.nasa.gov/xanadu/xanadu.html

34

whether correlation existed at higher frequencies between the VHE and optical bands;

no significant correlation was observed at least at short timescales.

10 4 10 3

10 2

10 1

100

101

Powe

r

VERITAS > 420 GeV PSD (original)

10 4 10 3

10 2

10 1

100

101

VERITAS > 420 GeV PSD (detrend)

10 4 10 3

Frequency (Hz)

10 2

10 1

100

101

Powe

r

Optical R-band PSD (original)

10 4 10 3

Frequency (Hz)

10 2

10 1

100

101

Optical R-band PSD (detrend)

Figure 11. VERITAS > 420 GeV (top) and optical R-band (bottom) PSDs with theoriginal observed light curves (left) and with light curves after long-term trend removal(right).

B. MODIFIED AUTO-CORRELATION ANALYSIS

The modified cross-correlation function (MCCF ) is defined as a function of the lag

time τ , with τ = k∆t:

MCCF (τ) =∑i

(x1(i∆t)− x1)(x2(i∆t+ τ)− x2)

σ1σ2

, (B1)

where x(t) is the number of counts in time bin (t, t + ∆t), and σ is the standard

deviation of x. For this modified correlation function, however, τ is not constrained

to be an integer multiple of the light curve bin size, ∆t, and can be incremented by

the time resolution element, δt. The MCCF can then be calculated for lag times,

τ = mδt (with m = 0,±1,±2, . . .) for light curves with a given timescale ∆t. The

timescale corresponding to the maximum of MCCF (k δt)/MCCF (0) gives the lag

time between x1 and x2.

From the definition of MCCF , the modified autocorrelation function (MACF ) is

obtained by setting x1 = x2,

MACF (τ) =∑i

(x(i∆t)− x)(x(i∆t+ τ)− x)

σ2. (B2)

The FWHM (full-width at half maximum) of the MACF is a measurement of the

variability duration. The maximum of FWHMMACF/∆t may be treated as a charac-

teristic timescale for the time series.

35

The MACF has the advantage over the regular ACF of being sensitive to variations

on time scales smaller than the typical time scales within which significant excess is

detected, potentially reaching the time resolution of the instrument δt,VER. Thus the

MACF method does not use an ad-hoc time binning (for example the 2-minute bins

from Figure 10). Rather, the MACF of the VERITAS flux was constructed assuming

a time resolution based on the minimum trigger rate of VERITAS during the Mrk 421

flare observations, δt,VER ∼ 0.7 ms. Then, for a range of ∆T ’s, where ∆T is some

integral number of δt,VER, the correlation was calculated between the initial time

series and a new series shifted by the time resolution δt,VER. This “sliding window”

process allows the MACF to find any characteristic variability on timescales shorter

than the time step ∆T .

Here we have applied the MACF to the night of the VHE flare (Epoch 3; see Sec-

tion 4.1) using all events above an energy threshold E = 420 GeV. Figure 12 shows the

FWHMMACF/∆T as a function of timescale where any characteristic timing signa-

ture would show up as a well-defined peak in the data points. MACFs from simulated

events following a pink noise process (with f−1.75 as above) were also calculated for

1000 iterations and are shown with the color map; they are bounded by dashed red

lines at the 99% confidence level.

C. X-RAY HYSTERESIS SEARCH

We analyzed the RXTE data during the VHE decline (Epochs 3-6), calculating

the hardness ratio between the 5–15 keV and 3–5 keV X-ray bands in 10-minute

time bins for each of the decline epochs. We then plotted the hardness ratio as

a function of counts in the combined 3-15 keV band to look for any evidence of

hysteresis. This is shown in the top panels of Figure 13 with a “zoom-in” shown in the

bottom panels corresponding to noticeable rise/decline states in the X-ray (indicated

by the solid/colored bands in Figure 6). All epochs show considerable evolution of

the hardness ratio with flux, with a variety of loops and trends. In particular, the top

left panel of Figure 13 with observations from Epoch 4 shows an apparent clockwise

hysteresis loop between count rates of 100–120 s−1 and hardness ratios of 0.02–0.06.

A zoom-in of part of this loop in the bottom left panel of Figure 13 (corresponding to

a burst-like feature in the light curve) shows a hardness ratio increase with increasing

count rate, followed by a slight, continued increase in hardness ratio as the count rate

begins to decrease, and finally a fairly constant hardness ratio even as the count rate

decreases significantly.

36

10 20 30 40 50Timescale [ s ]

0.6

0.8

1.0

1.2

1.4FW

HM /

Tim

esca

le3 confidence interval

0.0

0.1

0.2

0.3

0.4

0.5

MAC

F Fr

actio

n

Figure 12. Modified autocorrelation function for the VERITAS events above 420 GeV.Purple points show the MACF from VERITAS observations, while the color map boundedby red curves shows the region encompassed by MACFs from events simulated from a pinknoise process.

37

80 100 120 140 160

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18H

ardn

ess

Rat

io

80 100 120 140 160Count Rate ( s 1 )

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

80 100 120 140 160

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

90 95 100 105 110 115

0.03

0.04

0.05

0.06

0.07

Har

dnes

s R

atio

90 95 100 105Count Rate ( s 1 )

0.07

0.08

0.09

0.10

0.11

0.12

0.13

0.14

87.5 90.0 92.5 95.0 97.5

0.10

0.11

0.12

0.13

0.14

Figure 13. Top panels: Hardness ratio (between 5-15 keV and 3-5 keV) versus count rateof the RXTE observations of Mrk 421 over the three decline epochs indicated by the arrowsin Figure 6 (not strictly simultaneous with VHE data within the epoch). (a) Figure on left:Epoch 4; (b) figure in middle: Epoch 5 ; (c) figure on right: Epoch 6. The color shadesof the data points represent the chronological progression of the bursts, with lighter colorscorresponding to earlier times. Bottom panels: Zoom in on hardness ratio versus countrate of the RXTE observations of Mrk 421 corresponding to the colored bands in Figure 6(not strictly simultaneous with VHE data within the epoch). (a) Figure on left: Epoch 4zoom-in; (b) figure in middle: Epoch 5 zoom-in; (c) figure on right: Epoch 6 zoom-in. Thecolor shades of the data points represent the chronological progression of the bursts, withlighter colors corresponding to earlier times.

38

This research is supported by grants from the U.S. Department of Energy Office of

Science, the U.S. National Science Foundation and the Smithsonian Institution, and

by NSERC in Canada. This research used resources provided by the Open Science

Grid, which is supported by the National Science Foundation and the U.S. Depart-

ment of Energy’s Office of Science, and resources of the National Energy Research

Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science

User Facility operated under Contract No. DE-AC02-05CH11231. We acknowledge

the excellent work of the technical support staff at the Fred Lawrence Whipple Ob-

servatory and at the collaborating institutions in the construction and operation of

the instrument.

The MAGIC collaboration would like to thank the Instituto de Astrofısica de Ca-

narias for the excellent working conditions at the Observatorio del Roque de los

Muchachos in La Palma. The financial support of the German BMBF and MPG,

the Italian INFN and INAF, the Swiss National Fund SNF, the ERDF under the

Spanish MINECO (FPA2015-69818-P, FPA2012-36668, FPA2015-68378-P, FPA2015-

69210-C6-2-R, FPA2015-69210-C6-4-R, FPA2015-69210-C6-6-R, AYA2015-71042-P,

AYA2016-76012-C3-1-P, ESP2015-71662-C2-2-P, FPA201790566REDC), the Indian

Department of Atomic Energy, the Japanese JSPS and MEXT and the Bulgarian Min-

istry of Education and Science, National RI Roadmap Project DO1-153/28.08.2018

is gratefully acknowledged. This work was also supported by the Spanish Centro

de Excelencia “Severo Ochoa” SEV-2016-0588 and SEV-2015-0548, and Unidad de

Excelencia “Marıa de Maeztu” MDM-2014-0369, by the Croatian Science Foundation

(HrZZ) Project IP-2016-06-9782 and the University of Rijeka Project 13.12.1.3.02,

by the DFG Collaborative Research Centers SFB823/C4 and SFB876/C3, the Polish

National Research Centre grant UMO-2016/22/M/ST9/00382 and by the Brazilian

MCTIC, CNPq and FAPERJ.

This publication makes use of data obtained at Metsahovi Radio Observatory,

operated by Aalto University, Finland, and also the OVRO 40-m monitoring pro-

gram, which is supported in part by NASA grants NNX08AW31G, NNX11A043G

and NNX14AQ89G, and NSF grants AST-0808050 and AST-1109911. The UMRAO

data was obtained through NSF grant AST 0607523 and NASA Fermi GI award

NNX09AU16G. St.Petersburg University team acknowledges support from Russian

RFBR foundation, grant 12-02-00452. The Abastumani Observatory team acknowl-

edges financial support by the Shota Rustaveli National Science Foundation through

project FR/577/6-320/13. The Steward Observatory data were obtained under the

Fermi Guest Investigator Program grant NNX09AU10G. The research at Boston Uni-

versity is supported by NASA grant 80NSSC17K0649 and NSF grant AST-1615796.

39

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